All-pass matrices, the positive real lemma, and unit canonical correlations between future and past
type: Item
instance of: scholarly article
title: All-pass matrices, the positive real lemma, and unit canonical correlations between future and past
zbMATH Open document ID: 0664.62095
DOI: 10.1016/0047-259X(88)90108-X
author: S. H. Smith
published in: Journal of Multivariate Analysis
publication date: 1988-01-01T00:00:00Z
review text: Stationary vector ARMA processes x(t), \(t=0\), \(\pm 1\), \(\pm 2,...\), that are full rank are considered. It is shown, with E(z) denoting the `symbol' of the canonical correlation operator, that the number of zeros (counting multiplicities) on \(| z| =1\) of the transfer function, V(z), from innovations to outputs \(=\) (change in arg det E(z) around \(| z| =1)/2\pi =the\) number of unit Hankel singular values of E(s). It follows that the number of unit canonical correlations between the future x(t), \(t>0\), and the past x(t), \(t\leq 0\), is equal to the number of zeros of V(z) on \(| z| =1\), which is the result of \textit{E. J. Hannan} and \textit{D. S. Poskitt} [Ann. Stat. 16, No.2, 784-790 (1988; Zbl 0646.62082)]. This note, however, proves this result directly from the properties of E(z), which Hannan and Poskitt suggested should be possible. It is also shown that the result can be obtained using an entirely state space approach via the positive real lemma.
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 93E99
zbMATH DE Number: 4086825
zbMATH Keywords: all-pass matrices
zbMATH Keywords: Stationary vector ARMA processes
zbMATH Keywords: full rank
zbMATH Keywords: canonical correlation operator
zbMATH Keywords: number of zeros
zbMATH Keywords: number of unit Hankel singular values
zbMATH Keywords: number of unit canonical correlations
zbMATH Keywords: state space approach
zbMATH Keywords: positive real lemma
MaRDI profile type: Publication
Wikidata QID: Q125017136
cites work: Markovian Representation of Stochastic Processes by Canonical Variables
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cites work: All optimal Hankel-norm approximations of linear multivariable systems and theirL,∞-error bounds†
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cites work: Unit canonical correlations between future and past
cites work: Power spectrum reduction by optimal Hankel norm approximation of the phase of the outer spectral factor
full work available at URL: 0047-259x(88)90108-x
OpenAlex ID: W1971308366
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description: scientific article; zbMATH DE number 4086825